Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

The double bubble problem on the flat two-torus


Authors: Joseph Corneli, Paul Holt, George Lee, Nicholas Leger, Eric Schoenfeld and Benjamin Steinhurst
Journal: Trans. Amer. Math. Soc. 356 (2004), 3769-3820
MSC (2000): Primary 53A10; Secondary 49Q10
DOI: https://doi.org/10.1090/S0002-9947-04-03551-2
Published electronically: March 12, 2004
MathSciNet review: 2055754
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We characterize the perimeter-minimizing double bubbles on all flat two-tori and, as corollaries, on the flat infinite cylinder and the flat infinite strip with free boundary. Specifically, we show that there are five distinct types of minimizers on flat two-tori, depending on the areas to be enclosed.


References [Enhancements On Off] (What's this?)

  • [CCWB] Miguel Carrión Álvarez, Joseph Corneli, Geneveive Walsh, and Shabnam Beheshti, Double bubbles in the three-torus, Exp. Math., 12 (2003), 79-89.
  • [FT1] László Fejes-Tóth, Lagerungen in der Ebene auf der Kugel und im Raum, in Die Grundlehren der Math. Wiss., Vol. 65, Springer-Verlag, Berlin, 1953. MR 15:248b
  • [FT2] -, Regular Figures, A Pergamon Press Book, Macmillan Co., New York, 1964. MR 29:2705
  • [FT3] -, Über das kürzeste Kurvennetz, das eine Kugeloberfläche in flächengleiche konvexe Teile zerlegt, Math. Naturwiss. Anz. Ungar. Akad. Wiss., 62 (1943), 349-354. MR 9:460d
  • [F] Joel Foisy, Manual Alfaro, Jeffrey Brock, Nickelous Hodges, and Jason Zimba, The standard double soap bubble in $\mathbf{R}^{2}$uniquely minimizes perimeter, Pac. J. Math., 159 (1993), 47-59. MR 94b:53019
  • [H] Thomas C. Hales, The honeycomb conjecture, Discr. Comput. Geom., 25 (2001), 1-22. MR 2002a:52020
  • [HHM] Hugh Howards, Michael Hutchings, and Frank Morgan, The isoperimetric problem on surfaces, Amer. Math. Monthly, 106 (1999), 430-439. MR 2000i:52027
  • [HMRR] Michael Hutchings, Frank Morgan, Manuel Ritoré, and Antonio Ros, Proof of the double bubble conjecture, Ann. Math., 155 (2002), 459-489. Research announcement of same title, Electron. Res. Announc. Amer. Math. Soc., 6 (2000), 45-49. MR 2003c:53013
  • [Ma1] Joseph D. Masters, The double bubble on the flat torus, Unpublished notes on file with F. Morgan, Williams College, 1994.
  • [Ma2] -, The perimeter-minimizing enclosure of two areas in $\mathbf{S}^2$, Real Analysis Exchange, 22 (1996/1997), 645-654. MR 99a:52010
  • [M1] Frank Morgan, Geometric Measure Theory: A Beginner's Guide, 3rd ed., Academic Press, San Diego, 2000. MR 2001j:49001
  • [M2] -, Soap bubbles in $\mathbf{R}^2$ and in surfaces, Pac. J. Math., 165 (2) (1994), 347-361. MR 96a:58064
  • [MW] Frank Morgan and Wacharin Wichiramala, The standard double bubble is the unique stable double bubble in $\mathbf{R}^2$, Proc. AMS, 130 (2002), 2745-2751. Available on the web at http://www.ams.org/journal-getitem?pii=S0002-9939-02-06640-6. MR 2003c:53016
  • [RHLS] Ben W. Reichardt, Cory Heilmann, Yuan Y. Lai, and Anita Spielman, Proof of the double bubble conjecture in $\mathbf{R}^4$ and certain higher dimensional cases, Pac. J. Math., 208 (2003), 347-366.
  • [W] Wacharin Wichiramala, The planar triple bubble problem, Ph.D. thesis, University of Illinois at Urbana-Champaign, 2002.

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 53A10, 49Q10

Retrieve articles in all journals with MSC (2000): 53A10, 49Q10


Additional Information

Joseph Corneli
Affiliation: C/O Frank Morgan, Department of Mathematics and Statistics, Williams College, Williamstown, Massachusetts 01267 – and – Department of Mathematics, University of Texas, Austin, Texas 78712
Email: Frank.Morgan@williams.edu, jcorneli@math.utexas.edu

Paul Holt
Affiliation: Department of Mathematics and Statistics, Williams College, Williamstown, Massachusetts 01267
Email: pholt@wso.williams.edu

George Lee
Affiliation: Department of Mathematics, Harvard University, Cambridge, Massachusetts 02138
Email: lee43@fas.harvard.edu

Nicholas Leger
Affiliation: Department of Mathematics, University of Texas, Austin, Texas 78712
Email: nickleger@mail.utexas.edu

Eric Schoenfeld
Affiliation: Department of Mathematics and Statistics, Williams College, Williamstown, Massachusetts 01267
Email: eschoenf@wso.williams.edu

Benjamin Steinhurst
Affiliation: Department of Mathematics and Statistics, Williams College, Williamstown, Massachusetts 01267
Email: Benjamin.A.Steinhurst@williams.edu

DOI: https://doi.org/10.1090/S0002-9947-04-03551-2
Received by editor(s): June 16, 2003
Published electronically: March 12, 2004
Article copyright: © Copyright 2004 American Mathematical Society

American Mathematical Society